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Applying the ratio test

  1. May 22, 2015 #1
    Below is a screen shot of a solution to a problem. The part I don't fathom is after the ratio test is applied to the denominator. How can, noting that an+1, (2n-1) become (2n-1)(2n+1) and not just (2(n+1)-1)=2n+1?

    Thank you in advance
    Screenshot_NormalAppImage[0].png
     
  2. jcsd
  3. May 22, 2015 #2

    Mark44

    Staff: Mentor

    In an, the denominator is the product ##1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n - 1)##. What will be the next factor in the denominator for an + 1?

    BTW, when you post a question here, please don't delete the three parts of the homework template.
     
  4. May 22, 2015 #3
    Thanks for the quick reply

    Asked in that manner, the next part is 2n+1. But why not just replace the n with (n+1)? And why would the answer be completely different if this approach to problem is taken?
     
  5. May 22, 2015 #4

    Mark44

    Staff: Mentor

    It wouldn't. If you replace n by n + 1 in the expression 2n - 1, what do you get?
     
  6. May 22, 2015 #5
    You get 2n+1 instead of the (2n-1)(2n+1), no?
     
  7. May 22, 2015 #6

    Mark44

    Staff: Mentor

    You get (2n + 1). The (2n - 1) factor is the one from an.
     
  8. May 22, 2015 #7
    Thanks,
    I'm sorry but I still don't fathom.

    Let's say

    αn=2n-1 then αn+1 should equal 2(n+1)-1=2n+1

    Why the extra 2n-1 in the αn+1?
     
  9. May 22, 2015 #8

    Mark44

    Staff: Mentor

    This is not an, at least as it's defined in post 1.

    $$a_n = \frac{x^n}{1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n - 1)}$$

    Now, what is an + 1?
    You need to ask yourself how many factors are in the denominator of an? How many are in the denominator of an + 1?
     
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